Standard MCMC convergence diagnostics (such as the Gelman-Rubin statistic)
cannot be implemented in applications of GWMCMC because the parallel
chains of the Goodman-Weare method are not independent. Instead, the
standard approach to assessing convergence is to estimate the integrated
autocorrelation time, \(\tau _{\text{{corr}}}\) , of the chains, which
can be used to quantify the variance of the sampler. This can be leveraged
to determine how many steps are necessary to reduce the relative error
of the estimated MLE to be suitably small[1]},[2]}.
Specifically, the criteria for convergence is
\(z_{\gamma }\sqrt{{\frac{{\sigma _{\text{{$\theta $}}}^{2}}}{M_{\text{{eff}}}}}}\le \hat{{\mu }}_{\theta }\beta ,\) 
